PARALLELOGRAMS

Book I. Propositions 33 and 34

WE ARE MAKING OUR final approach to the theorem of Pythagoras. But we first have to establish when figures that are not congruent will be equal. To do this, we will look at quadrilaterals whose opposite sides are parallel. Such a figure is called a parallelogram. (Definition 15.) The following proposition effectively shows that such a figure exists.

The proposition proves that if two sides of a quadrilateral are equal and parallel, then the figure is a parallelogram. (Definition 14.) Hence we may construct a parallelogram; for, Proposition 31 shows how to construct a straight line parallel to a given straight line.

The next theorem has for its hypothesis that a figure is a parallelogram, that is, the opposite sides are parallel. And it proves what is obvious to the eye: the opposite sides are equal.

This theorem will be fundamental to the theory of equal areas. In fact, as we have pointed out several times, when we say that two figures are "equal," we mean that they are equal areas. See Problems 1 and 2 following Proposition 4.

The student should look at the figure below and, given that the figure is a parallelogram, it should be clear why those two triangles are equal.